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Light-Sheet Engineering Using the Field Synthesis Theorem

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Light-Sheet Engineering Using the Field Synthesis Theorem bioRxiv preprint doi: https://doi.org/10.1101/700633; this version posted July 13, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. Light-sheet engineering using the Field Synthesis theorem Authors Bo-Jui Chang1, and Reto Fiolka1,2. Affiliation 1 – Department of Cell Biology, UT Southwestern Medical Center, Dallas, TX, USA. 2 – Lyda Hill Department of Bioinformatics, UT Southwestern Medical Center, Dallas, TX, USA. Abstract Recent advances in light-sheet microscopy have enabled sensitive imaging with high spatiotemporal resolution. However, the creation of thin light-sheets for high axial resolution is challenging, as the thickness of the sheet, field of view and confinement of the excitation need to be carefully balanced. Some of the thinnest light-sheets created so far have found little practical use as they excite too much out-of-focus fluorescence. In contrast, the most commonly used light- sheet for subcellular imaging, the square lattice, has excellent excitation confinement at the cost of lower axial resolving power. Here we leverage the recently discovered Field Synthesis theorem to create light-sheets where thickness and illumination confinement can be continuously tuned. Explicitly, we scan a line beam across a portion of an annulus mask on the back focal plane of the illumination objective, which we call it C-light-sheets. We experimentally characterize these light-sheets and their application on biological samples. Keywords Field Synthesis, lattice light-sheet, C-light-sheet Introduction Light-sheet fluorescence microscopy (LSFM) has been transformative for volumetric imaging of single cells up to entire organisms, as it minimizes sample irradiation and allows efficient and rapid 3D imaging1-3. LSFM provides excellent spatiotemporal resolution, but is commonly not considered a super-resolution technique, as its spatial resolving power is still diffraction limited. Nevertheless, some LSFM implementations have been developed that can attain 300nm scale axial resolution3-7, effectively doubling the resolution of confocal microscopy, the workhorse in 3D microscopy. This is enabled by the larger set of angles that the separate illumination and detection objectives cover compared to a single objective microscope system. As such LSFM can improve the z-resolution over epi-fluorescence microscopes without compromising temporal resolution or requiring specialized fluorophores or non-linear optical phenomena. A driver for z-resolution in light-sheet microscopy is the thickness of the sheet, which in turn is governed by the laws of diffraction. For Gaussian beams, the thickness of the sheet is coupled via beam divergence to the confocal parameter of the beam waist, which dictates over what propagation distance a light-sheet can be approximated. While in principle Gaussian light- bioRxiv preprint doi: https://doi.org/10.1101/700633; this version posted July 13, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. sheets with sub-micron thickness can be created, their confocal parameter is usually shorter than a typical cell, which makes them ill-suited for practical imaging. Propagation invariant beams, such as Bessel8 and Airy9, 10 beams, and optical lattices, can in principle overcome the divergence of Gaussian beams. To create a light-sheet, such a beam is rapidly scanned laterally, a process known as digitally scanned light-sheet (DSLM)2. However, when propagation invariant beams are used for DSLM, for a given light-sheet thickness, any increase in confocal parameter is paid by reduced confinement of the excitation energy into the sheet. This has been first discovered with Bessel beam light-sheets11-14: while the main lobe of such a light-sheet is very narrow, it is accompanied by a large beam skirt that may contain over 90% of the energy within the light-sheet6. Thus large amounts of out-of-focus light are excited, which generate unwanted out-of-focus fluorescence and lead to accelerated photo-bleaching. To address this problem, the Betzig lab introduced lattice light-sheet microscopy (LLSM)3, which generates a light-sheet by coherent superposition of many Bessel beams. By carefully adjusting the spacing between the individual beams, optical lattices can be generated that can balance the thickness and the confinement of the light-sheet. The most widely used sheet in LLSM is the square lattice, which offers excellent confinement, but consists of a much thicker main lobe than a Bessel beam light-sheet. For higher axial resolution, the hexagonal lattice was proposed, which has a thinner main lobe than the square lattice, however at the cost of much stronger sidelobes. So far hexagonal lattice light-sheets have been rarely used, presumably because the sidelobes are difficult to remove numerically. Recently, we have discovered a new mathematical theorem called Field Synthesis that can be leveraged to create light-sheets in a more flexible and general way15. In Field Synthesis, a line is scanned over a mask conjugate to the back pupil of the illumination objective and a light-sheet is generated as an incoherent summation of the resulting instantaneous intensity distributions (see also Figure 1). While we have shown previously that this method can be used to recreate existing light-sheets faithfully, we explore here the potential of Field Synthesis to generate new light-sheets that are tailored to specific applications. In particular, our goal was to create light- sheets that have properties in between hexagonal and square lattices. Here we present experimental results of these new light-sheets that we have termed C-sheets, which are characterized in transmission and with fluorescent nanospheres, and are applied to biological imaging. Concept: Light-sheet design For the studies here, we restricted ourselves to an annular shaped pupil mask. The design rationales are outlined for light-sheet generation using the Field Synthesis approach (Figure 1), where a line is scanned over a pupil filter in Fourier space. As shown in Figure 2a, when analyzing the image formation process for one particular scan position of this line, the Fourier components of the light-sheet are given by the autocorrelation of the pupil function, in this case resulting in three line segments. These components give rise in real space to intensity distributions as shown in figure 1b, a sinusoidal pattern with a compact support. We note that a purely sinusoidal pattern would only result for an infinitely thin annulus mask. Such illumination pattern have recently been described as Bessel beams16, 17, however, mathematically they are Cosine patterns bound by an envelope. For simplicity, we will call such a beam a Cosine-Gauss beam. Using such a Cosine-Gauss beam as a light-sheet, we can compute the overall LSFM optical transfer function (OTF), which is the convolution of the Fourier components of the illumination and the wide-field optical transfer function (Figure 2b). As it can be seen, there are gaps bioRxiv preprint doi: https://doi.org/10.1101/700633; this version posted July 13, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. between the main three lobes of the LSFM OTF. This has been previously described in standing wave microscopy18, where a sinusoidal illumination pattern is created by interfering two laser beams from two opposing objectives. A remedy was readily found by Lanni and later Gustafsson by using multiple standing waves of varying axial frequencies to fill the gaps19, 20. This can be done for light-sheet illumination as well. If we scan a line over the full annulus (Figure 2c), a continuum of Cosine-Gauss beams is superimposed that contain the lowest spatial frequency (edges of the annulus) up to the highest spatial frequency that the optical system can produce (center of the annulus). As we have previously shown, the resulting time- averaged intensity distribution corresponds to a Bessel beam light-sheet15 (Figure 2d). If we instead use Field Synthesis to create a square lattice light-sheet, the beam is stepped to three discrete positions in the pupil, two at the edges of the annulus and one at its center (Figure 2c). The resulting light-sheet has significantly reduced ringing, but its main lobe is thicker than a Bessel beam light-sheet. For a hexagonal lattice the two main orders are shifted more inwards to the center of the annulus, and hence create a light-sheet with higher spatial frequency content. Through simulations we found that the central order is much weaker and does not contribute significantly to the final light-sheet (supplementary note). Thus, in an approximation, one can describe the light-sheet as a single Cosine-Gauss beam. Such a light-sheet features a thin main lobe, at the cost of stronger sidelobes compared to a square lattice sheet (Figure 2e). Imaging with strong sidelobes has been shown to be a challenge in other imaging modalities, such as 4pi21 and I5M20, and the removal of the resulting ghost images is difficult22. As a practical rule of thumb, the sidelobe strength should be kept below 50% (for a theoretical derivation, see Nagorni and Hell23). This is the main reason that 4pi microscopy has only been widely used with two-photon excitation, which suppresses the sidelobes to a tolerable level24. We hypothesized that we could create light-sheets with advantageous properties if we superimposed the spatial frequency range that exists between a square and a hexagonal lattice light-sheet. A continuum of intermediate frequencies can be generated if we scan a line from the position of the main order of the square lattice to the corresponding position of the hexagonal lattice (see Figure 2c). We expected that this would lead to a reduction in sidelobe strength (Figure 2e) and also would fill gaps in the OTF.
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